Retract problem and a subset admitting complete Boolean algebra structure of monoidally distributive posets

In [3], Hovey and Palmieri proved many interesing results around the Bousfield lattice of the stable homotopy caregory of spectra. In [4], the author, Shimomura and Tatehara defined monoidally distributive posets as a generalization of the Bousfield lattice. In this paper, we consider the retract problem and a subset admitting complete Boolean algebra structure of monoidally distributive posets. In the last section, we see that some results in [3] are given by our results in this paper.